Modeling group dispersal of particles with a spatiotemporal point - - PowerPoint PPT Presentation

Modeling group dispersal of particles with a spatiotemporal point process

Samuel Soubeyrand INRA – Biostatistics and Spatial Processes Joint work with L. Roques, J. Coville and J. Fayard 9th SSIAB Workshop, May 10, 2012

Group Dispersal Project – Plant Health and Environment Dpt.

Spatiotemporal point processes in propagation models

Object of interest: species spreading using small particles (spores, pollens, seeds...) Sources of particles generate a spatially structured rain of particles

◮ rain of particles → spatial point process ◮ spatial structure → inhomogeneous intensity of the process

Intensity of the spatial point process formed by the deposit locations of the particles

The intensity is a convolution between

◮ the source process (spatial pattern and strengths) and ◮ a parametric dispersal kernel

Simulation of an epidemics

Dispersal kernel

Dispersal kernel: probability density function of the deposit location of a particle released at the origin The shape of the kernel is a major topic in dispersal studies: it determines

◮ the propagation speed ◮ the spatial structure of the population ◮ the genetic structure of the population

Main characteristics of dispersal kernels:

◮ long distance dispersal (Minogue, 1989) ◮ non-monotonicity (Stoyan and Wagner, 2001) ◮ anisotropy

Abscissa (m) Ordinate (m) −200 −100 50 100 −300 −200 −100 100 200 Intensity 0.001 0.01 0.1 0.5

Observation of secondary foci (clusters) in real epidemics

Epidemics of yellow rust of wheat in an experimental field (I. Sache)

t = 1 t = 2 t = 3 t = 4

◮ Classical justifications for patterns with multiple foci:

◮ long distance dispersal ◮ spatial heterogeneity ◮ super-spreaders (a few individuals which infects many

susceptible individuals)

◮ Classical justifications for patterns with multiple foci:

◮ long distance dispersal ◮ spatial heterogeneity ◮ super-spreaders (a few individuals which infects many

susceptible individuals)

◮ An other justification to be investigated: Group dispersal

◮ Groups of particles are released due to wind gusts ◮ Particles of any group are transported in an expanding air

volume

◮ At a given stopping time, particles of any group are projected

to the ground

Group Dispersal Model (GDM): Spatial case

Deposit equation for particles: A single point source of particles located at the origin of R2 J: number of groups of particles released by the source Nj: number of particles in group j ∈ {1, . . . , J} Xjn: deposit location of the nth particle of group j satisfying Xjn = Xj + Bjn(ν||Xj||), (1) where Xj: final location of the center of group j, Bjn: Brownian motion describing the relative movement of the nth particle in group j with respect to the group center ν: positive parameter

Assumptions about the deposit equation

◮ The random variables J, Nj, Xj and the random processes

{Bjn : n = 1, . . . , Nj} are mutually independent

◮ Number of groups: J ∼ Poisson(λ) ◮ Number of particles in group j: Nj ∼indep pµ,σ2(·) ◮ Group center location: Xj ∼indep fXj(·)

(features of fXj: decrease at the origin is more or less steep, tail more or less heavy, shape more or less anisotropic...)

◮ The Brownian motions Bjn are centered, independent and

with independent components They are stopped at time t = ν||Xj||. Then, Bjn(ν||Xj||) ∼indep N(0, ν||Xj||I)

Dispersal from a single source

◮ Simulations: (Interpretation: Cox process or Neyman-Scott

with double nonstationarity — in the center pattern and the

• ffspring diffusion)

Abscissa Ordinate D e n s i t y Abscissa Ordinate

Abscissa Ordinate I n t e n s i t y −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 Abscissa Ordinate

Marginal probability density function (dispersal kernel):

Dispersal from a single source

◮ Simulations: (Interpretation: Cox process or Neyman-Scott

with double nonstationarity — in the center pattern and the

• ffspring diffusion)

Abscissa Ordinate D e n s i t y Abscissa Ordinate

◮ Marginal probability density function (dispersal kernel):

fXjn(x) =

• R2 fXjn|Xj(x | y)fXj(y)dy =
• R2 φν,y(x)fXj (y)dy.

The particles are n.i.i.d. from this p.d.f. while in the classical dispersal models the particles are i.i.d. from a dispersal kernel which may be of the form of fXj or fXjn

Discrepancies from independent dispersal

The GDM is compared with two independent dispersal models (IDM)

◮ IDM1: the number of particles in each group is assumed to be

• ne. Thus, particles are independently drawn under the p.d.f.

fXjn.

◮ IDM2: the number of particles in each group is assumed to be

• ne and the Brownian motions are deleted (i.e. ν = 0). Thus,

particles are independently drawn under the p.d.f. fXj.

Moments

X: Deposit location of a particle Q(x + dx): Count of points in x + dx

Criterion Model Value E(X) GDM ( 0

0 )

IDM1 ( 0

0 )

IDM2 ( 0

0 )

V (X) GDM V (Xj) + νE(||Xj||)I IDM1 V (Xj) + νE(||Xj||)I IDM2 V (Xj) E(||X||2) GDM E(||Xj||2) + 2νE(||Xj||) IDM1 E(||Xj||2) + 2νE(||Xj||) IDM2 E(||Xj||2) E{Q(x + dx)} GDM λµfXjn(x)dx IDM1 λfXjn(x)dx IDM2 λfXj (x)dx V {Q(x + dx)} GDM λ[µfXjn(x)dx + (σ2 + µ2 − µ)E{φν,Xj (x)2}(dx)2] IDM1 λfXjn(x)dx IDM2 λfXj (x)dx cov{Q(x1 + dx) GDM λ(σ2 + µ2 − µ)E{φν,Xj (x1)φν,Xj (x2)}(dx)2 , Q(x2 + dx)} IDM1 IDM2

GDM: larger variance of Q(x + dx) and positive covariance (decreasing with distance) → clusters (even with µ = 1) We expect multiple foci in the spatio-temporel case

Group dispersal model: Spatio-temporal case

GDM IDM1 IDM2

Abscissa Ordinate −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 Abscissa Ordinate −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 Abscissa Ordinate −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 Abscissa Ordinate 10000 20000 30000 40000 50000 60000 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 Abscissa Ordinate 10000 20000 30000 40000 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0 Abscissa Ordinate 5000 10000 15000 20000 25000 −1.0 −0.5 0.0 0.5 1.0 −1.0 −0.5 0.0 0.5 1.0

→ multiple foci under the GDM

Simulation study of the number of foci:

Definition

A δ-focus is a set of cells (from a regular grid) which are connected and whose intensity of points is larger than δ

50000 100000 150000 2 4 6 8

Threshold δ Number of −foci δ model ν σ

2

IDM2 IDM1 0.005 GDM 0.005 10 GDM 0.005 50 GDM 0.005 100 50000 100000 150000 2 4 6 8

Threshold δ Number of −foci δ model ν σ

2

GDM 0.001 50 GDM 0.005 50 GDM 0.01 50 GDM 0.1 50

Farthest particle (link with propagation speed)

Definition

The maximum dispersal distance during one generation is Rmax = max{Rjn : j ∈ J, n ∈ Nj} where Rjn = ||Xjn|| J = {1, . . . , J} if J > 0 and the empty set otherwise Nj = {1, . . . , Nj} if Nj > 0 and the empty set otherwise By convention, if no particle is released (J = 0 or Nj = 0 for all j), then Rmax = 0

Abscissa Ordinate

Rmax = max{Rjn : j ∈ J, n ∈ Nj} Under the GDM and IDMs, the distribution of the distance between the origin and the furthest deposited propagule is zero-inflated and satisfies: P(Rmax = 0) = exp

• λ{pµ,σ2(0) − 1}
• fRmax(r) = λfRmax

j

(r) exp{λ(FRmax

j

(r) − 1)}, ∀r > 0, where fRmax

j

is the p.d.f. of the distance Rmax

j

= max{Rjn : n ∈ Nj} between the origin and the furthest deposited propagule of group j, and FRmax

j

is the corresponding cumulative distribution function (FRmax

j

(r) = P(Rmax

j

= 0) + r

0 fRmax

j

(u)du). → Distribution of Rmax

j

?

Under the IDMs, Nj = 1 for all j ∈ J and, consequently, pµ,σ2(0) = 0 and fRmax

j

(r) = fRjn(r) = 2π rfXjn((r cos θ, r sin θ))dθ for the IDM1 2π rfXj((r cos θ, r sin θ))dθ for the IDM2.

Under the GDM, the distribution of Rmax

j

is zero-inflated and satisfies: P(Rmax

j

= 0) = pµ,σ2(0) fRmax

j

(r) =

• R2 fRmax

j

|Xj(r | x)fXj(x)dx

=

+∞

• q=1

qpµ,σ2(q)

• R2 fRjn|Xj(r | x)FRjn|Xj(r | x)q−1fXj(x)dx,

where fRjn|Xj is the conditional distribution of Rjn given Xj satisfying fRjn|Xj(r | x) = 2r r2 h1(u, x)h2(r 2 − u, x)du, hi(u, x) = fi(√u, x) + fi(−√u, x) 2√u , ∀i ∈ {1, 2}, fi(v, x) = 1

• 2πν||x||

exp

• −(v − x(i))2

2ν||x||

• ,

∀i ∈ {1, 2}, x = (x(1), x(2)) and FRjn|Xj(r | x) = r

0 fRjn|Xj(s | x)ds.

Theorem

Consider a GDM and an IDM1 characterized by the same parameter values except that E(J) = ˜ λ, E(Nj) = ˜ µ and V (Nj) = σ2 for the GDM, and E(J) = ˜ λ˜ µ, E(Nj) = 1 and V (Nj) = 0 for the IDM1 (⇒ same marginal dispersal kernel). Then, for all r > 0 the probability P(Rmax ≥ r) is lower for the GDM than for the IDM1.

Theorem

Consider an IDM1 and an IDM2 characterized by the same parameter values except that ν > 0 for the IDM1 and ν = 0 for the IDM2. Then, for all r > 0 the probability P(Rmax ≥ r) is lower for the IDM2 than for the IDM1. Interpretation: The population of particles are less concentrated in probability for the IDM1 than for the GDM and the IDM2

E(Rmax):

0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

ν Expected maximum distance

IDM1 IDM2 GDM ( =10) σ2 GDM ( =50) σ2 GDM ( =100) σ2

0.00 0.02 0.04 0.06 0.08 0.10 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

ν Expected maximum distance

IDM1 IDM2 GDM ( =10) σ2 GDM ( =50) σ2 GDM ( =100) σ2

Conclusion

◮ With group dispersal, one can generate multiple foci whereas

the particles are more concentrated

Perspectives

◮ Toward analytic results about the farthest particle in the

spatio-temporal case (→ speed of propagation of epidemics)

◮ Inference (with Tomas Mrkvicka and Eyoub Sidi) ◮ Alternative representations of group dispersal (Cylinder-based

models, with Tomas Mrkvicka and Antti Penttinen)

◮ Study of the evolutionary dynamics between group dispersal

and independent dispersal